Forward speed motions of a submerged body. The Cauchy problem. (English) Zbl 0568.35059

The linearized problem for the motion of a submerged body in a perfect, inviscid, irrotational fluid, occupying an unbounded domain of \(R^ 3\), moving forward with a forced motion depending on (x,y) but not z, is \(\Delta u=0\), \(u|_{\Sigma}=\phi\), \(\partial u/\partial n|_{\Gamma}=f(t,\cdot)\), and \(\phi_ t-a\cdot \phi_ x+w\cdot \phi_ y=-b\eta\), \(\eta_ t-a\cdot \eta_ x+(w\cdot \eta)_ y=du/\partial n|_{\Sigma},\) \(\phi |_{t=0}=\phi_ 0\), \(\eta |_{t=0}=\eta_ 0\). In this paper the elliptic and the initial- value problem are viewed as coupled by a pseudo-differential operator T. The Yosida approximation \(T^{\epsilon}\) of T leads to a regularized initial-value problem. The perturbation theory of nonlinear semigroups is used to obtain weak solutions of the problem as \(\epsilon\) goes to zero.
Reviewer: G.F.Webb


35L45 Initial value problems for first-order hyperbolic systems
35J25 Boundary value problems for second-order elliptic equations
47H20 Semigroups of nonlinear operators
35D05 Existence of generalized solutions of PDE (MSC2000)
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[1] Beale, Eigenfunction expansion for objects floating in an open sea, Comm. Pure Appl. Math. XXX pp 283– (1977) · Zbl 0336.76004
[2] Friedman, The initial value problem for the linearized equations of water waves, J. of Math. and Mechan. 17 (2) pp 107– (1967)
[3] Garipov, On the linear theory of gravity waver: the theorem of existence and uniqueness, Arch. rat. Mech. Anal. 24 pp 352– (1967) · Zbl 0149.45603
[4] Hanouzet, Espace de Sobolev avec poids, application au problème de Dirichlet dans un demi-espace, Rend. della sem. Math. Univ. di Padova 46 pp 227– (1971)
[5] Jami , A. Etude théorique et numérique de phénomènes transitoires en hydrodynamicque navale
[6] Kato , T. Perturbation theory for linear operators Springer Verlag 132 · Zbl 0148.12601
[7] Lau, The linearized equation of water waves, Indiana Univ. Math. J. 22 (3) pp 233– (1972) · Zbl 0228.35054
[8] Lions , J. L. Quelques méthodes de résolution de problèmes aux limites non linéaires 1969
[9] Lions, Problèmes aux limites non homogènes 1 (1968) · Zbl 0235.65074
[10] Licht , C. Etude de quelques modèles décrivant les vibrations d’une structure élastique dans la mer · Zbl 0574.73063
[11] Nedelec, Approximation des équations intégrales en mécanique et en physique (1977)
[12] Newman, The theory of ship Motion, Adv. in Appl. Mech. 18 pp 221– (1978)
[13] Sclavounos , P. 1983
[14] Tartar , L. Topics in non linear analysis 78 13
[15] Treves, Introduction to pseudodifferential and Fourier integral Operators 1 (1980)
[16] Wendland, On the integral equation method for the plane mixed, Math. Meth. Appl. Sci. 1 pp 265– (1979) · Zbl 0461.65082
[17] Wilcox , C. H. Scattering theory for the d’Alembert equation in exterior domains 1975 · Zbl 0299.35002
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